J^u(k) = $ d^4x J^u(x) exp[-i p_s x^s]    (1)

Does this change at all in curved spacetime?  In particular, is it
necessary to generalize the Fourier Kernel exp[-i p_s x^s], to something
else, for example:

exp[-i p_s x^s] --> exp[-i p_s x^s + F(x)]   (2)

where F(x) is some function of the coordinates x^u?

Or, does (1) still suffice?

Thanks,

   Richard Melrose's lecture notes are available here :
   <http://www-math.mit.edu/~rbm/iml90.pdf>

Depending on the nice properties of the transform you want to preserve
and depending on the context (e.g., replacing R^n by a Riemannian
manifold), in many cases an analog of the Fourier transform does not
exist at all.

That is not to say that there are no useful generalization of the
Fourier transform; there are many. The kinds of generalizations that
people find useful are often grouped under the moniker of "harmonic
analysis". Here's a old post by John Baez that gives examples of some
of these generalizations:

news:7t8eac$abk@charity.ucr.edu
http://groups.google.com/group/sci.physics.research/msg/3b85eb9ddc0ad4e3

Would you please check out this link:

http://en.wikipedia.org/wiki/Pontrjagin_duality#Fourier_transform

Then, please advise whether in your view this approach is applicable to
curved spacetime, and, more generally, please advise what the
limitations may be on this approach, if any.

Also, am I correct to read this so as to say that xi and xi-bar are be
inverses, such that:

xi xi-bar = 1  ?

Thanks,

In general, finding the Green functions for a wave equation in a
curved space-time is hard and there is no magic wand like "a
generalized Fourier transform" that you can wave to quickly find them.
If you look at any paper on QFT on a specific space-time, a large part
of the work is always devoted to finding the appropriate Green
function, using every available trick from the book, exactly because
there is no one method that always works.

It seems that the requirement that curved spacetime with metric g_uv
must have Minkowski space as a tangent space, i.e. with vierbein V:

g^\mu\nu = V^mu_a V^nu_a eta_ab   (1)

may qualify many of the curved, smooth, simply-connected spacetime
manifolds of general relativity as locally compact, and they are
certainly Abelian.  Is it right to think that "locally compact" =
"locally Euclidean," and if not, what does such a view either include
that it should not or exclude that it should not?

Also, was I correct to read
http://en.wikipedia.org/wiki/Pontrjagin_duality#Fourier_transform so as
to say that xi and xi-bar are be inverses, such that:

xi xi-bar = 1  ?   (2)

Thanks,

More importantly, let me go back to to what Baez said in the link you
gave earlier:

"Some of the nice stuff works just because R^n is a Lie group
- this stuff is called harmonic analysis on Lie groups.  A
good example is how you can decompose any function on a
compact Lie group like SU(n) or SO(n) into a linear
combination of matrix elements of irreducible representations.
This is called the Peter-Weyl theorem. . . .

The idea here is that even if your manifold doesn't have much
symmetry at all, it still looks *locally* like R^n, so you can
do a kind of local analogue of Fourier analysis on it.

Basically, the more symmetry your space has, the easier
it is to do something like Fourier analysis on it.  Above
I listed 4 of the main branches of harmonic analysis, in
order of decreasing symmetry."

It seems to me like Baez is suggesting that a curved spacetime manifold
which is locally Minkowski "still looks *locally* like R^n," (here, R^4)
and so we can do "a kind of local analogue of Fourier analysis on it."

And, it seems that he is saying we can do a Fourier-analog analysis on
the curved space, so long as it possesses requisite symmetry.  So, the
question then becomes, "what symmetries are required?"

It also looks like if we restrict ourselves to Lie Groups on a curved
manifold with is locally Minkowskian and use a suitable Haar measure,
that we can get some generalization of Fourier analysis to work there.

And, finally, of course two points displaced on a manifold cannot be
dealt with as if they were in a Euclidean vector space.  But, does not
the whole parallel transport analysis which underlies the curvature
tensor R^u_abv supplant and generalize the ability to deal with vectors
in a consistent way, on a curved manifold?

Thanks,

Thus one can get closed form Green functions for any sufficiently
symmetric space-time, not only for Minkowski space (which has a
particularly simple - abelian - transitive symmetry group, whose
representation theory is determined by the ordinary Fourier transform).

But, of course, this doesn't help much for studying general
general relativity.

But, as I read the Baez post at
http://groups.google.com/group/sci.physics.research/msg/3b85eb9ddc0ad4e3
which Igor recommended, I am of the impression that the question is one
of what subset manifolds from among all possible
mathematically-permitted curved spacetime manifolds which might
otherwise be permitted by general relativity, have sufficient symmetry
to allow the derivation of closed form Green functions.  Are you saying
that there is *no curved manifold at all* for which this will work, or
that any curved space for which this will work must have certain
symmetries, and, if the latter, then the question becomes, "what are the
requisite symmetries?"

If there is *no curved manifold at all* for which this can be done, then
that would seem to be saying that path integral quantization only works
for a flat spacetime background, and that we need to find some other
foundation for quantum field theory if we wish to reconcile quantum
theory with gravitation.  If on the other hand, there is some subset of
manifolds for which this works, then perhaps what this means is that the
path integral formulation remains valid in curved spacetime, but in the
process forces the elimination of certain curved manifolds from
consideration which do not have the requisite symmetry.  Given that
physics is a process of elimination of many mathematical possibilities
which are not physically permitted down to those select few which are
physically-permissible, the restriction to manifolds with certain
symmetries that do permit closed form Green function derivation may not
be a bad thing at all, and may in fact be driving us toward what can be
physically real while winnowing out that which cannot be.

Your thoughts?

Thanks.

I should emphasize again that QFT in curved space-times is a mature
field described in books and review papers. If you want to learn about
it, you should look them up and read them. Fixating on non-existent or
unuseful generalizations of Fourier transforms is much less
productive.

But for a group representation approach, it is enough to have a
homogneous space (still a highly symmetric space but  less than a
symmetric space), and there are infinitely many of these even in 4D
(one just needs 4 independent Killing fields), some of them of
high interest to cosmology.

On the other hand, the less symmetries there are the more difficult
is the analysis, and only the symmetric space case is fully developped.

I haven't followed the literature on this closely, so can't give
references. But Volume 1 of Thirring's treaatise on math physics
gives a classification of highly symmetric space-times.

The corresponding transformations for path integrals get additional
determinants from the transformations, whcih are not well understood
rigorously (but informally handled with ghost fields).

Thus all this is just open territory, not a clsed road with valid
no-go theorems.

Because of the big bang, there cannot be time invariance, and
realistic cosmological models with symmetry only have 3 independent
Killing fields.

First, let me say that I appreciate your dialogue with both me and Igor.
It has been very helpful in trying to clarify the issues involved with
doing path integral quantization, which implicitly requires some form of
harmonic analysis, in curved spacetime.  I think a separate thread might
be also suitable, titled "Does, and if so under what conditions does,
path integration quantization apply to curved spacetime?"

On the supposition that path integration does and should apply to curved
spacetime, or at least to a symmetry-restricted subset of curved
spacetime manifolds, I have posted an exercise paper at
http://jayryablon.files.wordpress.com/2009/11/path-integration-of-the....
If you can be so kind as to briefly take a look at this, it would be
much appreciated.

In the the first three sections 1-3 I calculate the QED Green functions
in curved spacetime, with boundary terms included not discarded.  This
can be done fully and successfully, but only up to the point of finding
an explicit expression for the propagator.

To make further progress on an explicit expression for the propagator,
it seems unavoidable that one must of necessity do some type of
"Fourier" analysis analog, which we all have been calling "harmonic
analysis," in curved spacetime.  I am glad to hear that you regard this
as "open territory."  I am hoping that sections 4 through 7 might help
to better define that territory.

*If nothing else, I would ask you to please look at section 4.*  Section
4 mirrors the discussions we have been having here, and even quotes some
of the very helpful statements you have provide in this thread.  I
expect that this is a "first draft" of whatever it eventually becomes,
but I would like to know if I am at least talking basic sense in this
section.  After some extended discussion of the issues in this thread, I
arrive in section 4 at the point where I progress to calculations of the
path integral by using Pontryagin duality, and thereby accept whatever
restrictions are placed on spacetime manifolds when one uses that
particular analysis technique which -- form what I can tell -- is the
closest we can get to Fourier analysis.

In section 5, I then attempt to how that gauge symmetry itself, greatly
facilitates the ability to conduct harmonic analysis in curved
spacetime, and does allow exact calculations to be done with Pontryagin
duality.  By equation (5.34), I develop forward and inverse Pontryagin
duality transformations, which differ from but can be expressed in terms
of ordinary Fourier transformations, and which allow for explicit
calculation of the propagator in any curved spacetime manifold that has
the requisite symmetries to make Pontryagin duality an admissible
technique.

Section 6 veers off into Yang-Mills theory, which seems to emerge
naturally out of the gauge symmetry analysis in Section 5.

In section 7, I finally get to an explicit Pontryagin duality
calculation of the propagator in the spacetime manifold under
consideration, which, by definition, must be a manifold to which
Pontryagin duality applies.

Once again, I very much appreciate the information you have provided and
the discussions so far, and look forward to further discussion and
exploration of this "open territory."

Thanks,

On the handwaving level (i.e., in the way how all path integral stuff
is currently dealt with in physics), the path integral in curved
space is already well understood, and people work with this for a
long time.

Thus further progress in understanding the path integral must
come from understanding the flat case first, in a nonperturbative,
rigorous fashion.

I think this may be just a question of my not using the right
terminology.

However one divides up the terminology between "Green function" and
"propagator," the calculation of the path integral has two parts to it
and that is the way in which I was using different terminologies for
each part.  Specifically:

First, one must calculate path integral Z as a series expansion of the
form:

Z = sum G^(s)  (1/n!) J^n  (1)

where G^(s) includes what I refer to as the "propagator" D(x-y) written
as a function of (x-y) in spacetime and not yet as a function of the
momentum variable of integration (p) in momentum space.  An example of
this part of the overall calculation is (3.39) in
http://jayryablon.files.wordpress.com/2009/11/path-integration-of-the....

In this particular case of (3.39), it happens that one only has even
powers of J, and I have kept one power of D(x-y) grouped with the J^2
and the multiple powers of D(x-y) inside G^(s).  This is what I refer to
as "calculating the Green functions" though perhaps it is better
referred to as calculating the series expansion for the Green functions
or some such thing.

Secondly, one must obtain an explicit expression for D(p) rather than
D(x-y), which I call the "spacetime propagator" and the "momentum space
propagator" respectively, and that requires us to have some way to go
back and forth between D(x-y) <--> D(p) and in flat spacetime we use
Fourier analysis but in curved spacetime we need something else that we
gave been talking about as "harmonic analysis."  That is what I address
starting in section 4.

I do not think there is anything wring in my calculation of (3.39) which
builds on everything up to that point, but only in the words I am using
for this.  Perhaps you can make clear what the right language is in
which to discuss this.

On this question, may I refer you to (5.34) of
http://jayryablon.files.wordpress.com/2009/11/path-integration-of-the...,
which are the transformations I use to go between D(x-y) <--> D(p).
Clearly, there is Fourier analysis involved but not the same Fourier
analysis which occurs in Minkowski spacetime because one has the
sqrt(-g) and the Fourier(sqrt(-g(x))) and delta^(4)(p) in the
transformations.  Thus, we are using compositions -- actually
convolutions which contain sqrt(-g), see (5.28) -- which do not show up
in Minkowski space.  And so, I am entertaining the idea that what
happens in curved spacetime are not the same Fourier transformations
which happen in flat spacetime, but do continue to employ the same
Fourier analysis, where the sqrt(-g) gets convolved with the functions
of (x-y) and (p) we are transforming.  What may be an key question is
whether (5.34) in fact preserves the mapping between x <--> p spaces in
the way that is required, and I do not know how to rigorously assess
this point.  Here, I have in mind Arnold's earlier point that " As long
as the space-time is diffeomorphic to a homogeneous space one can use a
diffeomorphism to transform coordinates to that space, then do the
Fourier analysis there, then transform back. "

Now, I do in (5.34) use the same kernel exp[-ipx] as in flat spacetime,
and that may and probably will strike you on first impression as wrong.
In general non-Euclidean geometry, this would indeed be wrong.  But, in
physics, we also have gauge symmetry, and I believe that gauge symmetry
permits one to do this, because one can always "gauge out" certain
arbitrariness from the kernel including the arbitrariness of a general
coordinate transformation, in gauge theory.  This is a central part of
my argument, and the purpose of my section 5 is to explain why I believe
this is so.

Thanks,

The relevant theorem is one which equates the space C^{infinity}(M, C)
of functions on a manifold M to the complex numbers C to a C*-algebra
and -- conversely -- identifies a commutative C*-algebra as a function
space over a manifold.

The correspondence, roughly speaking, is x <-> delta_{x} where delta_x
is the (singular) function which yields integrals
   integral_M f(y) delta_x(y) dy = f(x).
More precisely, the map
   delta_x: f |-> f(x)
yields a "minimal" homomorphism from the function space C^{infinity}
(M, C) to each point x in M. In the converse direction, a commutative
C*-algebra A has a family of minimal 1-dimensional ideals, A_x = C,
each represented by a map d_x: A -> A_x = C, which is linear d_x(f +
g) = d_x(f) + d_x(g) and preserves products d_x(fg) = d_x(f) d_x(g).

All this comes under the header of the GNS correspondence.

The exercise to follow may or may not be related to the above theorem.
It is as follows.

In field theory, one has an issue integrating over momentum space.
This shows up most prevalently when doing the Wick time-ordered
operator expansion for a term that represents a loop diagram.

The traditional approach is to yank out an ax and chop off the space
at some radius |p| = Lambda.

Bear in mind, as this is done, that the space is endowed with an IN-
definite metric. So, what the "scale cut off" actually is, is NOT a
cut off of "small scale" (as you'll read in almost all textbook
presentations and hear of in almost all explanations of what' s
happening here), but a cut-off away from the light cone p^2 = 0.

So, now the exercise is this: replace the cut-off with a
compactification of momentum space -- a redefinition of momentum space
onto a curved compact geometry.

Bear in mind that momenta are not things to be considered in
isolation. Fundamental systems in particle and field theories have
state spaces that are irreducible representations of the underlying
space-time symmetry group. This means, the Lie algebra (and more
generally, the Poisson-Lie manifold) for the momentum generators
   {P_{mu}, P_{nu}} = 0
is part of a larger structure that also contains the generators {J_{mu
nu} = -J_{nu mu}} for rotations and boosts.

There are two ways to go with this:
(1) something along the lines, {P_{mu}, P_{nu}} = lambda J_{mu nu}
or
(2) a "Penrose" compactification of the P's, themselves, into a
teleparallel geometry (which, here, means a curved geometry that is
the group manifold of a non-Abelian Lie group), with the J's combined
with the P's via a semi-direct product.

In both cases, the subtlety is how the compactification will work with
the fact that the metric is indefinite. In case (2), you're talking
about a compactification of Minkowski space (e.g. Penrose's
compactification into a projective space).